Abstract
We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.
Highlights
As remarked in [1, p. 121], elliptic operators with variable coefficients naturally arise in several areas of physics and engineering
We will look for the solution of the Dirichlet problem for the operator E in the domain Ω in the form of a simple layer potential
Where s(z, x) and sk(z, x) denote the fundamental solution for Laplace equation and the double k-form associated with s(z, x), respectively
Summary
As remarked in [1, p. 121], elliptic operators with variable coefficients naturally arise in several areas of physics and engineering. ⋅ δp1⋅1⋅⋅⋅⋅m⋅pkq1⋅⋅⋅qm−k L (x, y) dxq1 ⋅ ⋅ ⋅ dxqm−k dyp1 ⋅ ⋅ ⋅ dypk = τm−k,k (x, y) + (−1)k(m−k) ∗yλm−k (x, y) , where τm−k,k(x, y) satisfies (20) on account of (as1j1 ⋅ ⋅ ⋅ askjk ) (x) − (as1j1 ⋅ ⋅ ⋅ askjk ) (y) = O (x − y) (27) ∂L (x, ∂xs y) as1 ⋅⋅⋅sk i1 ⋅⋅⋅ik (y) δ1⋅⋅⋅m jj1 ⋅⋅⋅jk ik+2 ⋅⋅⋅im Following Fichera we employ the parametrix L to construct a principal fundamental solution of the differential operator E (see [12]).
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